New method tackles continuous-variable problems on qubit-based machines

Researchers at the Veneto Institute of Oncology IOV-IRCCS in Padova, Italy, are addressing a core challenge in quantum optimization: efficiently tackling continuous-variable problems without being limited by either predefined search ranges or the need for complex Hamiltonians. The team’s new framework combines adaptive discretization with an iterative mapping onto an effective spin-glass Hamiltonian, offering a flexible approach for qubit-based quantum architectures. The method was assessed using the n-dimensional Gramacy-Lee benchmark and applied to an inverse-planning problem for intensity-modulated radiotherapy, demonstrating the potential for competitive solutions in both quantum and quantum-inspired pipelines.

Adaptive Discretization for Continuous Variable Encoding

Constructing a usable Hamiltonian for continuous-variable quantum optimization often demands either dense polynomial models or complex interpolation schemes, limiting practical application. The framework detailed by researchers avoids these requirements by employing an adaptive discretization technique, where the resolution of each continuous variable’s encoding adjusts dynamically during the optimization process. This approach differs from standard binary discretization, which confines encoded values to a fixed domain and resolution, and is a departure from methods that relax constraints at the expense of resource efficiency.

The new method associates each continuous variable with a small qubit register, the size of which is determined by a variable-dependent resolution parameter. This allows the encoding to adapt to the optimization landscape, unlike previous binary encodings restricted to a predefined interval; a binary encoding restricts a variable to the interval where K is a normalization constant.

The team demonstrates this adaptation through a discretization scheme, illustrated by a representation of dynamically-adapted discretization, showing how possible values of a continuous variable are encoded using bit strings around a reference value. Each variable is assigned its own discretization parameter, controlling the range of encoded values around that reference point. Solutions obtained using this adaptive discretization strategy, even with only two qubits per variable, closely matched those from a classical benchmark on the original continuous objective function.

Data presented by the researchers further illustrates the range of values explored during optimization for each initial discretization parameter, highlighting the method’s ability to efficiently search the solution space. This approach offers a flexible mapping strategy for continuous-variable optimization in both qubit-based quantum and quantum-inspired pipelines, potentially broadening the scope of solvable problems.

Local Surrogate Models & Spin-Glass Hamiltonian Construction

The framework detailed in this work constructs a surrogate Hamiltonian around the current best configuration at each iteration, a process that relies on generating a set of neighboring configurations to evaluate the objective function. This approach differs from directly optimizing the original continuous objective function, allowing for a local approximation that can be processed by quantum or quantum-inspired solvers. The number of neighboring configurations generated, denoted as M, is specifically chosen to match the number of unknown coefficients within the surrogate Ising model, a detail important for maintaining accuracy and computational efficiency.

For a fully-connected model, this number scales with the number of qubits used in the encoding. The resulting surrogate Hamiltonian is quadratic in spin variables, enabling optimization via a tree tensor network (TTN) ansatz and a DMRG-like variational ground-state search.

This method was tested against both Best Neighbor (BN) and Simulated Annealing (SA) algorithms on the unconstrained n-dimensional Gramacy-Lee benchmark, providing a comparative analysis of optimization strategies. The team also evaluated the approach on a constrained variant of the benchmark and a proof-of-concept inverse-planning problem in fixed-gantry intensity-modulated radiotherapy (IMRT) for cancer treatment, demonstrating its versatility across different problem types. Comparisons with Particle Swarm (PS) optimization offer an external benchmark, assessing performance against a standard classical optimizer operating directly within the continuous search space.

The researchers found that the TTN-based implementation consistently outperformed the Best Neighbor baseline, and showed favorable results when compared to Simulated Annealing on the same discretized representation. The ability to map the objective function onto an effective spin-glass Hamiltonian, independently of its analytic form, is a key feature of this method, offering flexibility in addressing complex optimization problems.

Tree Tensor Network Optimization via DMRG-like Search

The optimization process relies on a tree tensor network (TTN) ansatz, a variational representation of quantum states where complex calculations are broken down into smaller, interconnected tensors. This decomposition allows for a DMRG-like search for the ground state of a surrogate Hamiltonian, effectively simplifying the optimization problem for qubit-based solvers. Once the surrogate Hamiltonian’s low-energy state is prepared, candidate solutions are extracted by sampling bit-strings using tensor-network techniques.

A key advantage of this approach lies in its solver-agnostic stopping criterion, which focuses on the convergence of the objective function rather than a fixed number of iterations. This is particularly important when comparing different optimization algorithms, as the meaning of an iteration can vary significantly between them. Analysis of the optimization runs, conducted over multiple independent trials, reveals the performance characteristics of each solver.

The team specifically examined the impact of qubit count per variable on the n-dimensional Gramacy-Lee benchmark, using both two and eight qubits. These comparisons highlight the TTN-based solver’s ability to navigate the surrogate Hamiltonian efficiently, while PS is applied directly to the original continuous objective function.

Gramacy-Lee Benchmark for Unconstrained Optimization

The research team evaluated their framework using the n-dimensional Gramacy-Lee function, a standard benchmark for continuous optimization due to its non-convex and oscillatory nature. Initial results demonstrated that both the tree tensor network (TTN)-based solver and SA outperformed BN, effectively escaping local minima and identifying superior solutions. A key distinction in the evaluation involved Particle Swarm (PS) optimization, which operated directly on the original continuous objective function, providing an external benchmark against a conventional classical optimizer.

This comparison probed the relative effectiveness of different optimization strategies on the encoded problem versus direct optimization in the native continuous search space. The team’s work highlights that TTN consistently achieved the best overall performance on the Gramacy-Lee benchmark, suggesting an efficient exploration of the solution space. Further analysis extended to a constrained variant of the Gramacy-Lee function, incorporating penalty terms to test the framework’s ability to handle optimization problems with limitations.

Constrained Gramacy-Lee Function Testing with Penalties

This extension assessed how effectively the adaptive discretization and local spin-glass mapping could navigate problems with defined boundaries and restrictions, a common requirement in real-world applications. The Gramacy-Lee function, with its non-convex and oscillatory characteristics, presents a demanding test for any continuous optimization method, particularly those aiming for qubit-based quantum architectures. Its structure intentionally avoids the straightforward binary encoding used in standard QUBO/PUBO formulations, forcing a more sophisticated approach to problem representation.

A comparison of the TTN-based implementation with Best Neighbor, Simulated Annealing, and Particle Swarm optimization revealed distinct performance characteristics across different algorithms. The team observed that TTN consistently outperformed both algorithms on this benchmark, suggesting an advantage in navigating complex, multi-dimensional landscapes. Further validation involved applying the framework to a proof-of-concept fixed-gantry intensity-modulated radiotherapy (IMRT) inverse-planning problem, where the objective function combined polynomial dose-matching terms with non-polynomial dose-constraint penalties.

IMRT Inverse-Planning as a Radiotherapy Application

The research team addressed a practical challenge within intensity-modulated radiotherapy (IMRT) by applying a new framework for continuous-variable optimization, a technique used in cancer treatment planning. This approach sidesteps limitations inherent in fixed-grid discretizations, which restrict the scope of potential solutions by predefining search parameters. Instead, the method uses adaptive discretization, allowing for a more flexible exploration of the optimization space during IMRT inverse-planning.

This application presented a non-polynomial objective function, a combination of polynomial dose-matching terms and non-polynomial dose-constraint penalties, that does not easily translate into a low-order binary encoding suitable for many quantum algorithms. The framework’s capacity to handle this complexity demonstrates its potential to address realistic constraints encountered in clinical treatment planning.

The researchers assessed the approach using the n-dimensional Gramacy-Lee benchmark, alongside both unconstrained and constrained instances, to test the incorporation of penalty terms. Particle Swarm optimization acted directly on the original continuous variables, while Simulated Annealing operated on the same discretized surrogate Hamiltonian as the TTN-based solver. The results indicate that the adaptive discretization effectively avoids premature convergence, a common pitfall in complex optimization problems.

The team acknowledges that these results are a proof of principle, intended to demonstrate the flexibility of the encoding and surrogate-construction strategy rather than clinical-scale performance. Future work will focus on scaling the method to larger IMRT instances, optimizing tensor-network strategies and exploring alternative surrogate solvers. These developments, the researchers suggest, would support applications to more advanced radiotherapy modalities such as Volumetric-Modulated Radiotherapy (VMAT) and Helical Tomotherapy. The team included members from Università degli Studi di Padova, CNRS, INFN and PlanQC GmbH, indicating a collaborative effort spanning academia and a German quantum technology company.

QUBO/PUBO Formulation Challenges for Continuous Problems

Constructing a Hamiltonian for continuous-variable optimization presents a specific hurdle: many objective functions exhibit oscillatory and non-polynomial behavior, complicating direct encoding into the low-order binary formats expected by standard Quadratic Unconstrained Binary Optimization (QUBO) or Polynomial Unconstrained Binary Optimization (PUBO) models. This limitation necessitates either restricting the search space through fixed-grid discretizations or developing more complex Hamiltonian representations, both of which introduce trade-offs in precision and computational cost.

The n-dimensional Gramacy-Lee benchmark function, with its continuum of values, provides a demanding test case for evaluating strategies to overcome this challenge. When employing a fixed number of qubits per variable, a standard binary-to-decimal conversion limits the representation to a discrete set of values, potentially sacrificing the nuances of the continuous objective. The framework detailed in this work addresses this by offering a flexible approach to representing continuous problems on discrete quantum resources, moving beyond the limitations of fixed discretization.

Comparison to Baseline Solvers: Best Neighbor & Simulated Annealing

The new optimization framework consistently outperformed a Best Neighbor baseline across multiple problem sizes tested using the n-dimensional Gramacy-Lee benchmark, demonstrating its ability to identify improved solutions more efficiently. These findings suggest that the adaptive discretization strategy employed by the framework contributes to its robustness and ability to navigate complex optimization landscapes. The framework’s ability to effectively handle both polynomial and non-polynomial terms is significant, as many real-world optimization problems exhibit such hybrid characteristics.

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